Surface Area of Cylinders

(1) Right Circular Cylinder

  1. Curved surface area = the perimeter x the height of the cylinder, i.e. S = 2\pi rh
  2. The area of each of the flat surfaces, i.e. of the ends,  = \pi {r^2}
  3. The total surface area  = 2\pi rh + 2\pi {r^2} = 2\pi r\left( {r + h} \right)

 

Example:

Find the height of the solid circular cylinder if the total surface area is 600 sq.cm and the radius is 5cm.

Solution:
            Here    r = 5cm
Total surface area  = 2\pi {r^2} + 2\pi rh = 660
Or        2\pi r\left( {r + h} \right) = 660
Or        2 \times \frac{{22}}{7} \times 5\left( {5 + h} \right) = 660
Or        5 + h = \frac{{660 \times 7}}{{5 \times 44}} = 21
\therefore           h = 16cm

 

Example:

A cylindrical vessel without a lid has to be coated on both its sides. If the radius of its base is \frac{1}{2}m and its height is 1.4m, calculate the cost of tin coating at the rate of $2.25 per 1000 sq.cm.

Solution:
Given that
Radius of the base of cylindrical vessel, r = \frac{1}{2}m = 50cm
Height,              h = 1.4m = 140m
\therefore Area to be tin coated  = 2\left( {{\text{curved surface + area of base}}} \right)
 = 2\left( {2\pi rh + \pi {r^2}} \right)
 = 2\pi r\left( {2h + r} \right)
 = 2 \times 3.14 \times 50\left( {2 \times 140 \times 50} \right) = 314 \times 330
 = 103620sq.cm

Now, the cost of tin coating per 1000sq.cm = $2.25
            \therefore Total cost of tin coating  = \frac{{2.25}}{{1000}} = 103620 = 233.15 dollars

 

(2) Hollow Circular Cylinder

  1. Curved surface area  = 2\pi Rh + 2\pi rh = 2\pi \left( {R + r} \right)h
  2. Total surface area  = 2\pi \left( {R + r} \right)h + 2\pi \left( {{R^2} - {r^2}} \right)

 

(3) Elliptic Cylinder

A cylinder with a base which is an ellipse is called an elliptic cylinder. If a and b are the semi-major axis and semi-minor axis and h is the height, then

  1. Volume  = \pi abh
  2. Curved surface area  = \pi \left( {a + b} \right)h
  3. Total surface area = \pi \left( {a + b} \right)h + 2\pi ab